Enclosing Circle Algorithm (Rework)
Contents
K-Means Based Algorithm
Location
The script and associated modules are located in:
E:\McNair\Software\CodeBase\New Implement of Enclosing Circle (Constrained K Means, Smallest Circle)\googleplotter.py
Explanation
The algorithm relies on an object oriented implementation of a "cluster".
Each "cluster" has the following associated with it:
1. area of minimum circle enclosing points in the cluster 2. points associated with the cluster 3. minimum circle associated with the cluster (x-coord,y-coord,radius) 4. "parent" cluster 5. "children" cluster(s)
Inputs:
1. n, the minimum number of points to include in a circle 2. k, the amount of clusters to split the points into 3. valid_nodes, an empty list to store the possible clusters to explore 4. explored_nodes, an empty list to store the explored clusters 5. cur_node, initialized to an object with all values set to none 6. end_nodes, initialized to an empty list used to store the clusters which cannot be split further
Algorithm:
1. Calculate the area of the minimum circle enclosing the total set of points 2. Split initial points into "k" clusters of at least "n" points using Constrained-K-Means Algorithm 3. construct minimum circles of "k" clusters and compute their total area 4. if this area <= area of the area of the first circle 5. Add all "k" clusters to "valid_nodes" 6. for node in valid_nodes, add node to explored_nodes, change cur_node to node, and update respective areas, parents, children, circle, points 7. run this algorithm again, change k = 2 8. if the cur_node cannot be split into 2, or the cur_node cannot be split into 2 circles with smaller areas than its parent, add cur_node to end_nodes and resume algorithm with next valid_node not in explored_node as the cur_node 9. if all the valid_nodes are in explored_nodes, return end_nodes 10. repeat the above algorithm for all k from 2 to int(#number of total points/n) and take the end_nodes with the lowest total area
Runtime:
The runtime of the minimum enclosing circle is O(n), and the runtime of constrained k-means depends on k itself. Here, the value of k is proportional to the total number of points.
We would expect the algorithm to slow as the number of points increase. For reference, a set of 80 points with n =3, took about 30 minutes.
Further improvements could implement an early-stopping method to converge to a local optima.
Visualization
The K-Means based algorithm returns the optimal solution (left), and faster.
Step By Step Use
1. Open up the file "googleplotter" 2. where it says with open("C:\....") replace the filepath in the parentheses with your filepath enclosed in quotes. 3. Press ctrl+s to save the file 4. open up the command line 5. type: python googleplotter.py 6. your resulting text file and html google plots will be outputted to: E:\McNair\Software\CodeBase\New Implement of Enclosing Circle (Constrained K Means, Smallest Circle)
Brute Force
Location
The Julia Script is located in:
E:\McNair\Software\CodeBase\Julia Code for enclosing circle
Explanation
1) Find all combinations of two points in the total data set.
2) For each of these combinations, draw a circle and figure out what other points are contained in a circle around those two points. If the circle contains at least n points, add it to a set of valid circles. There are Number of Points choose 2 possible circles.
3) Combine the valid circles in all possible ways and see if the resulting scheme contains all of the points. If it does, add it to a set of valid schemes. I believe that the number of ways to do this is the sum from i = 1 to i = number of valid circles Number of Circles choose i.
4) Iterate through the valid schemes and calculate the areas of the schemes.
5) Return the scheme with the minimum area.